Search results for "Finite element approximations"

showing 3 items of 3 documents

Functional A Posteriori Error Estimates for Time-Periodic Parabolic Optimal Control Problems

2015

This article is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of the functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds. peerReviewed

Mathematical optimizationControl and OptimizationMathematicsofComputing_NUMERICALANALYSISFinite element approximations010103 numerical & computational mathematicsType (model theory)01 natural sciencesparabolic time-periodic optimal control problemsError analysisFOS: MathematicsApplied mathematicsMathematics - Numerical AnalysisNumerical testsfunctional a posteriori error estimates0101 mathematicsMathematics - Optimization and Control49N20 35Q61 65M60 65F08Mathematicsta113Time periodicta111Numerical Analysis (math.NA)State (functional analysis)Optimal controlComputer Science Applications010101 applied mathematicsOptimization and Control (math.OC)multiharmonic finite element methodsSignal ProcessingA priori and a posterioriAnalysisNumerical Functional Analysis and Optimization
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Time Dependent Case

1999

This chapter is devoted to finite element approximations of scalar time dependent hemivariational inequalities. We start with the parabolic case following closely Miettinen and Haslinger, 1998. At the end of this chapter we discuss, how the results can be extended to constrained problems. Our presentation will follow the structure used for the static case in Chapter 3. First, we introduce an abstract formulation of a class of parabolic hemivariational inequalities (see Miettinen, 1996, Miettinen and Panagiotopoulos, 1999).

Scalar (mathematics)Applied mathematicsFinite element approximationsBilinear formFinite element methodMathematics
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Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation

2020

We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson-Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp., 69:481-500, 2000] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computa…

a priori error estimatesClass (set theory)Correctness010103 numerical & computational mathematics01 natural sciencesMeasure (mathematics)guaranteed and efficient a posteriori error boundsFOS: MathematicsApplied mathematicsPolygon meshMathematics - Numerical Analysis0101 mathematicserror indicators and adaptive mesh refinementMathematicsNumerical AnalysisApplied MathematicsRegular polygonNumerical Analysis (math.NA)convergence of finite element approximationsLipschitz continuity010101 applied mathematicsComputational MathematicsNonlinear systemexistence and uniqueness of solutionssemilinear partial differential equations65J15 49M29 65N15 65N30 65N50 35J20MathematikA priori and a posterioriPoisson-Boltzmann equationdifferentiaaliyhtälöt
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